Optimal. Leaf size=77 \[ \frac {\sqrt [4]{\cos ^2(a+b x)} \sqrt {d \sec (a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac {1}{4},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c d (m+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2586, 2577} \[ \frac {\sqrt [4]{\cos ^2(a+b x)} \sqrt {d \sec (a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac {1}{4},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2586
Rubi steps
\begin {align*} \int \frac {(c \sin (a+b x))^m}{\sqrt {d \sec (a+b x)}} \, dx &=\frac {\left (\sqrt {d \cos (a+b x)} \sqrt {d \sec (a+b x)}\right ) \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^m \, dx}{d^2}\\ &=\frac {\sqrt [4]{\cos ^2(a+b x)} \, _2F_1\left (\frac {1}{4},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) \sqrt {d \sec (a+b x)} (c \sin (a+b x))^{1+m}}{b c d (1+m)}\\ \end {align*}
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Mathematica [C] time = 1.75, size = 289, normalized size = 3.75 \[ \frac {8 c (m+3) \sin ^2\left (\frac {1}{2} (a+b x)\right ) \cos ^4\left (\frac {1}{2} (a+b x)\right ) F_1\left (\frac {m+1}{2};-\frac {1}{2},m+\frac {3}{2};\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (c \sin (a+b x))^{m-1}}{b (m+1) \sqrt {d \sec (a+b x)} \left ((\cos (a+b x)-1) \left ((2 m+3) F_1\left (\frac {m+3}{2};-\frac {1}{2},m+\frac {5}{2};\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+F_1\left (\frac {m+3}{2};\frac {1}{2},m+\frac {3}{2};\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right )+(m+3) (\cos (a+b x)+1) F_1\left (\frac {m+1}{2};-\frac {1}{2},m+\frac {3}{2};\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \sec \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{m}}{d \sec \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\sqrt {d \sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x +a \right )\right )^{m}}{\sqrt {d \sec \left (b x +a \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\sqrt {d \sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{\sqrt {\frac {d}{\cos \left (a+b\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin {\left (a + b x \right )}\right )^{m}}{\sqrt {d \sec {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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